Planning Radiation Treatment Capacity to Meet Patient Waiting Time Targets*

Transcription

1 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1 Planning Radiation Treatment Capacity to Meet Patient Waiting Time Targets* Siqiao Li, Na Geng, Member, IEEE, and Xiaolan Xie, Fellow, IEEE Abstract This paper proposes a queueing and patient pooling approach for radiotherapy capacity allocation with heterogeneous treatment machines called LINAC (Linear Accelerators), different waiting time targets (WTT) and treatment protocols. We first propose a novel queueing framework with LINAC time slots as servers. It leads simple single-class queues for evaluation of WTT satisfaction that would, otherwise, require analysis of complicated multi-class queues with re-entrance. Mixed integer programming models are given for capacity allocation and case-mix optimization under WTT constraints. We then extend the queueing framework by pooling basic patient types into groups sharing the same slot servers. A mathematical programming model and a pairwise merging heuristic are proposed for patient pooling optimization to minimize the overall LINAC capacity needed to meet all WTT requirements. Extended numerical experiments are conducted to assess the efficiency of our approach and to show properties of optimal capacity allocation and patient pooling. Index Terms Radiotherapy, capacity allocation, queueing, patient pooling, mathematical programming T I. INTRODUCTION HE automation community has shown increasing interests in healthcare engineering. Automation techniques such as discrete event simulation, queueing analysis and planning/scheduling have proven helpful in improving healthcare delivery. Meanwhile, specific features of healthcare setting bring new challenges that need novel modelling frameworks and approaches for design and operations of healthcare delivery systems. Motivated by our collaboration with the Ruijin hospital in Shanghai facing severe imbalance between capacity and demand for radiotherapy, this paper proposes a novel queueing-based approach for capacity planning and allocation of a stochastic discrete event dynamic system with the following challenging features. *This work is supported in part by the Natural Science Foundation of China ( , , ), in part by the specialized research fund for the doctoral program of higher education ( ), and in part by the Agence Nationale de Recherche (ANR-11-TECS ) (Corresponding author: Xiaolan Xie). All authors are with the Department of Industrial Engineering and Management, Shanghai Jiao Tong University, Shanghai, China. {aprilsiqiao, xie, Xiaolan Xie is also with Centre for Biomedical and Healthcare Engineering, CNRS UMR 6158 LIMOS-IEOR, Ecole Nationale Supérieure des Mines, Saint Etienne, France. xie@emse.fr Complex re-entrant treatment processes: radiotherapy is delivered not once, but in a number of consecutive sessions with breaks on weekends. Once a patient starts her first treatment, she will come back on the treatment machine called LINAC (Linear Accelerators) for a given number of consecutive working days. Treatment on the same LINAC is preferred to avoid extra calibration and location. Regularity of the treatment appointments is also needed to allow proper recovery. Diversity of patients with different waiting time targets (WTT) and treatment processes: LINACs are shared by patients with different treatment intent, urgent level, and cancer site. The number of sessions varies from 1 to 40 and the waiting time target from 48 hours to over a month. Heterogeneous treatment machines with different capabilities: Apart from basic treatment machines, there are more and more advanced ones such as RMCI, Cyberknife, Novalis and Gamma-Knife. They are all called LINACs in this paper for simplicity. LINACs differ by radiation energies and autonomous operation capabilities. Some LINAC can be used by nearly all patients, whereas others can only be used for basic radiotherapy treatments. Cancer is the first cause of mortality in urban China and the second cause in rural China with 220 million new cases and 160 million deaths each year. About 70% of cancer patients in China are expected to undergo radiotherapy treatment. Radiotherapy treatment is a complicated process including pre-treatment and treatment. Patients first go through a series of pre-treatment processes including consultation, examination, tumor location, and simulation and so on. Pre-treatment phase ends with radiotherapy treatment planning which specifies the number of treatment sessions, the dosage of rays, and the type of treatment machines called LINAC to use. The radiotherapy treatment starts when a corresponding LINAC is available and is delivered on daily consecutive sessions with breaks on weekends according to the patient s treatment plan [1-2]. LINACs are expensive treatment machines and the bottleneck of most radiotherapy services. Long waiting times from when the patient is ready to begin the treatment and its actual start are observed worldwide [3]. Long waiting time for radiotherapy has been shown [4] to negatively impact on the clinical outcome, cause patients anxiety, deteriorate the quality of their life and even compromise their chance of successful treatment. The risk of local recurrence increases with increasing waiting time for RT and therefore waiting times for RT should be as short as

2 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 2 reasonably achievable. Therefore, waiting time targets (WTT), i.e., the longest time between the ready-to-treat date and the treatment start so that tumors remain at a lower growth rate, have been recommended in the literature. WTT are 4 weeks in Denmark [3], 14 days in Canada for non-emergency and nonurgent cases [5], and 31 days in UK [6]. The waiting time targets tend to be diagnosis specific. In Denmark, patients with intestinal and head and neck cancer have to be treated without delays [3]. The Joint Council for Clinical Oncology in UK adjusted good practice and maximum acceptable waiting time to both treatment intent (palliative or radical) and waiting list status (emergency, urgent, routine) [6]. Unfortunately, significant deviations from these targets were observed. Fewer than 75% of the radiation therapy treatments in British Columbia are initiated within the 14-day target [5]. Our field observations in the Ruijin Hospital lead to similar findings. As the top ranked hospital in Shanghai, the radiotherapy department faces much higher demand than its capacity and patients have to wait up to two months for radiotherapy. The waiting time analysis shows that the limited LINACs capacity is the major cause of this lengthy waiting. The research questions addressed in this paper include then how much capacity is needed to meet patient waiting time targets, how to allocate the overall capacity among different patient groups and what is the optimal case-mix if the capacity is not enough. Apart from these strategic and tactical decisions, hospital managers can also improve patient waiting time at the operational level by appropriate real-time patient admission policies, an issue of our future research. Whether or how capacity allocation influences the system efficiency is one important but difficult issue. It is well known that capacity sharing among similar customers improves resource utilization. However, it is not always beneficial to pool multiple queues into one [7-9]. Pooling of different M/G/n queues with different service time but the same workload was show in [7] to be not beneficial for customers with lower service time in terms of expected waiting time and tail probability. Queues with FCFS and NPP (non-pre-emptive policy) policies were considered in [8]. It was shown that the optimal pooling partitions the queues into several groups and it depends on mean service time, arrival rate and coefficient variation of service times. Pooling two M/M/1 queues with different waiting time targets was shown not always beneficial [9]. We show in Section V.B that pooling queues does not always reduce the capacity requirement. The majority of the limited existing literature on radiotherapy operations management addresses patient planning and scheduling. Most studies on the treatment phase address patient scheduling [6, 10-14]. Heuristic algorithms were proposed in [6, 10] to schedule patients forward from the first feasible start date (ASAP algorithm) or backward from the last feasible start date (due date) in order to minimize the number of patients who break WTT. Two independent types of LINAC (high rays and low rays), each with a separate waiting list, were considered. Deterministic integer programming models were proposed in [11-13] to schedule patients in a given waiting list in order to maximize the number of scheduled patients without breaching waiting time target. Only one LINAC type was considered. A deterministic multi-objective mathematical model was proposed in [14] to schedule not only the date for each treatment session but also assign LINAC and specific slots of that day to patients considering various LINAC types. The objective is to minimize patients waiting time and the number of patients who break WTT. A stochastic model and an approximate dynamic programming approach were proposed in [5] for dynamic scheduling of radiotherapy patients and overtime capacity on one type of LINACs. The closely related work is radiotherapy patient booking [3]. Two patient types, urgent patients to be treated as soon as possible and others with longer waiting time targets, were considered. Based on historical data analysis, upper and lower booking limits were given each day for each type of patients. Restrictive assumptions include (i) identical LINACs and (ii) same number of treatment sessions and same session duration for patients with the same WTT. Rather than analytical results, this paper is only a statistical analysis. Capacity planning and allocation have also attracted some attention in other healthcare settings. Effective allocation of expensive imaging diagnosis capacity among several classes of patients within a day was addressed in [15]. A contractbased strategy was proposed in [16] for MRI examination reservation for stroke patients. An average cost Markov decision process approach was proposed to assign stroke patients to reserved capacity to reach the best balance of average waiting time and unused reservation. An M/G/ queue-based approach was proposed in [17] for periodic reallocation of beds to services in order to minimize the expected overflows. A newsboy model was proposed in [18] to schedule surgery orderings and surgery start times in order to best balance the idle time of operating rooms and waiting time of surgeons. We are not aware of existing literature on capacity planning and allocation with multiple WTTs. Our contributions are multifold. First, we propose a novel queueing framework with LINAC time slots as servers instead of the traditional LINAC servers. This novel queueing perspective allows elegant modeling of the re-entrance of a patient as her service time on a given slot server. It leads to easy evaluation of the LINAC capacity needed to meet the WTT requirement of each patient type, i.e. to ensure their WTT with a certain probability. Mixed-integer programming models for LINAC capacity allocation and case-mix optimization are proposed in order to provide each patient type with enough slot servers and hence to meet the WTT requirements. The queueing approach is then extended to pool patient types into groups served by the same slot servers in order to reduce the total number of slots needed to meet WTT requirements. A mathematical programming model and an efficient heuristic are proposed to patient pooling optimization. Given that queue pooling is not always beneficial for customers with different service capacity and service level requirements [7-9], our combinatorial optimization pooling approach is the first to take advantage of the scale effect of pooling subject to detailed service level constraints. Finally we

3 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 3 provide a complete sensitivity analysis to show properties of optimal capacity allocation and patient pooling. Note that some preliminary results appeared in a conference paper [19]. The remaining of this paper is organized as follows. Section II sets the LINAC capacity allocation problem. Section III proposes mathematical models for capacity allocation, patient pooling and case-mix optimization. Section IV presents solution techniques. Numerical experiments are conducted in Section V to assess the effectiveness of our solution techniques and to perform the sensitivity analysis of the optimal capacity allocation. Section VI concludes the paper. II. A NEW LINAC CAPACITY ALLOCATION FRAMEWORK A. LINAC servers vs slot servers Large radiotherapy departments treat different types of patients with different types of LINACs. Advanced LINACs can treat more types of patients than the basic ones. Every day, the set of patients on a LINAC is different.... P 9 P 8 P 7 P 6 Waiting List of Patients and allocation difficult and time consuming due to the blackbox nature of discrete event simulation. In this paper, we propose a totally different modeling approach for LINAC capacity allocation. The daily capacity of each LINAC is divided in some basic time slots corresponding to different session durations. Each time slot, e.g. slot 8:00-8:10 on LINAC 1, is considered as a server that we sometimes call slot server. The service time of a patient on a slot server is exactly the number of treatment sessions she needs. When a slot server finishes a patient, it will be released for a new patient having the same session duration. A LINAC working 8 hours a day with identical session durations of 10 minutes is then represented by 48 slot servers. LINAC Server Perspective LINAC Slot Server Perspective Waiting List of Patient Group 1 Waiting List of Patient Group 2 10min 10min 10min 5min 9:00 LINAC 1 LINAC 1 LINAC 1 5min Patient 5 Patient 5 Patient 4 Waiting List of Patient Group 3 15min Session duration (Slot) Patient 4 Patient 3 Patient 4 Patient 6 Patient 7 Patient 6 15min Waiting List of Patient Group 4 10min 8:00 Patient 2 Patient 2 Patient 1 Patient 1 Patient 1 Day 1 Day 2 Day 3 Fig. 1. Treatment of patients on a LINAC Due to patient arrivals/departures, different sets of patients are treated on the same LINAC on different days. Fig. 1 illustrates this situation with seven patients treated on a LINAC for three consecutive days from 8:00 to 9:00. Patient 6 starts her first session on day 2 as patient 3 completes on day 1. Patient 7 starts her first session as patient 5 and patient 2 complete on day 2. A straightforward model of such system is a multi-server multi-class queue with patients as customers and LINACs as servers. Unfortunately, such queueing systems are fairly complex due to specific features of radiotherapy. Each server is shared by different classes of patients with different session durations. Multiple treatment sessions of the same patient lead to complicated re-entrant queues that are difficult to analyze. LINACs have different capabilities. Different types of patients have different treatment plans, and require different numbers of treatment sessions on different LINACs with different session durations. Last but not the least different patients have different WTT for the start of their treatment. Given the above complexity, to the best of our knowledge, few existing studies in the literature all use discrete event simulation for performance evaluation of radiotherapy. No analytical analysis exists. This makes the capacity planning Fig. 2. Treatment process under slot server perspective Fig. 2 shows the treatment process under the slot server perspective. According to session durations of different patient classes, the capacity of a LINAC is split into different groups of slot servers. Patients of the same class wait in the same queue for corresponding slot servers. As a result, each patient class can be modeled a multi-server queue with the number of sessions as service time. This slot server modeling has some attractive features. The slot server concept agrees with the requirement that each patient receives all her radiotherapy treatment sessions without interruption on the same LINAC. It also agrees with the requirement of reducing appointment time deviations for different sessions of the same patient in order to allow proper recovery. The re-entrance phenomenon is avoided and the number of treatment sessions is modeled simply as a discrete service time on the same slot server. As a result the complicated re-entrant systems are transformed into simple queueing systems from the slot-server perspective. If each slot server is dedicated to a group of patients, the slot-server queueing system can be further decomposed into some single-class queues, each corresponding to a group of patients with the number of time slots dedicated to it. The assumption is reasonable for capacity allocation and leads to easy and independent evaluation of the waiting time targets.

4 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 4 B. Problem setting In this paper, patients are classified into some basic types according to the following attributes: the set of LINACs on which patients can be treated, the number of sessions, the session duration and WTT. In practice, patients are usually sorted into several types considering two dimensions: treatment intent and emergency. Patients of the same type have similar condition, similar treatment protocols, and the same WTT as a result of similar urgency. One of the most important criteria in radiotherapy is the service level. Service level indicators include average waiting time of patients [14, 16], the number of patients scheduled [11-13] and the deviation from WTT [6,10,14]. Our investigations of Chinese hospitals show that the tumors remain stable and there is no need to recalculate the treatment plan within the corresponding WTT. It is more significant for the hospital to reduce the number of patients breaching WTT than reduce average waiting time. The service level is measured in this paper by the percentage of patients meeting their WTT, i.e. the probability P(W i > ω i ) of the actual waiting time W i of a type-i patient exceeding its waiting time target ω i. The service level is said achieved if P(W i > ω i ) α i where α i is the maximal acceptable breaching probability of type-i patients. Different basic patient types can also be merged into a single patient group and their waiting queues pooled into a single queue in order to reduce LINAC capacity requirement while meeting all service level requirements. Throughout the paper, a patient group is a set of patient types. The following assumptions are made throughout the paper. A1. Patients of the same type have the same session number, the same WTT, the same set of LINACs that can treat them, and the same LINAC-dependent session duration. In reality, the first session is usually longer due to the setup of the LINAC, we simply average this extra workload among all treatment sessions and assume identical session duration. This is reasonable as the number of patients starting their radiotherapy is small with respect to the total number of patients treated on a day. A2. The daily capacity of LINACs is decomposed into slots with different durations. Each slot is assigned to a patient group of correct duration. Weekend breaks are neglected and the waiting time of a patient is measured in the number of working days she waits. A3. Different types of patients arrive independently each according to a Poisson distribution. Patients of the same group are treated according to some priority rule as its slot servers become available. Every patient remains on a slot server till all her treatment sessions complete. The priority rule is either FIFO or a static priority rule. A4. Balking and reneging of patients are not considered and each patient leaves only after the completion of all her treatment sessions. In reality, some patients may turn to other hospitals if the expected waiting time is too long. For simplicity, LINACs are assumed all different. Two identical LINACs are considered as LINACs of different types having the same attributes. The following assumptions are made for patient pooling. A5. Patients of the same group have the same set of LINACs that can treat them and the same session duration on each of these LINACs. This implies that two patient types cannot be merged in the same group if either they have different sets of LINACs or their session durations are different. This assumption is motivated by the observation of our field study that most patients have similar session durations. The latter is in line with the real data given in [5]. A6. The cost of pooling different patient types into one group is neglected. In reality, different patient types may come from different medical departments so that patient pooling may lead to integrated management and hence additional cost. However this additional cost is balanced by the scale economy of pooling. One salient feature of our patient pooling approach is the departure from the usual tempting aggregation approach which replaces grouped patient types with an artificial representative patient type. Such approach leads to approximation of exact WTT requirements and session numbers. Instead, in this paper, we do not aggregate merged patient types but only have them waiting in a single queue for pooled servers. Each patient type still keeps its own WTT requirements and number of sessions needed. The problem consists of (i) partitioning the set of patient types into patient groups and (ii) to assign slot servers from different LINACs to patient groups such that the detailed service level requirement of each patient type is achieved. If the total LINAC capacity is not enough, we also consider the case-mix optimization problem in order to determine the admission ratio of each patient type such that the total revenue is maximized subject to all service level requirements. III. MATHEMATICAL PROGRAMMING MODELS A. Common notation Patients attributes: I: set of patient types indexed by i = 1,2,, I ; J i : set of LINACs that can treat type-i patients, J i J ; ω i : WTT in days of type-i patients; α i : maximal breaching probability for type-i patients; λ i : daily arrival rate of type-i patients; l i : number of treatment sessions of type-i patients; t ij : session duration in minutes of type-i patients on LINAC j. LINACs attributes: J : set of LINAC type, indexed by j = 1,2,, J ; I j : set of patient types that LINAC j can treat, I j I; C j : daily capacity in minutes of LINAC j.

5 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 5 B. Capacity Allocation with given patient pooling Consider a given patient pooling partitioning the set I of patient types in patient groups and defined by the following: G : set of patient groups indexed by g = 1,2,, G ; I g : set of patient types in group g, I g I. Thanks to assumptions A2 and A3, once the capacity allocation decision is made, different patient groups become independent with each served by its dedicated slot servers. The following notation is well-defined and its evaluation addressed later: N g : minimal number of slot servers needed to meet all service level requirements of its patient types, i.e. WTT constraint (α i, ω i ) with daily arrival rate λ i for all patient type-i with i I g. Decision variables of the capacity allocation include: s gj : nonnegative integer variable denoting the number of slots of LINAC j assigned to group-g patients; γ: LINAC utilization ratio needed to meet all service level requirements. The capacity allocation problem (CAP) consists in allocating LINAC capacity to patient groups in order to best balance the overall LINAC utilization and satisfy all service level constraints. subject to min γ jεjg s gj N g, gεg (1) gεgj t gj s gj γc j, jεj (2) γ R, s gj N, gεg, jεj (3) where, thanks to assumption A5, J g, G j, and t gj are similar and well-defined notation as J i, I j and t ij. (1) ensures the WTT requirements of all patients are met. (2) is the capacity constraint of each LINAC. (3) are variable constraints. Once N g are determined, the capacity allocation model can be solved easily with standard mixed-integer programming solvers. A balanced LINAC capacity allocation s gj is derived and all service level requirements are met. γ 1 means that current capacity is enough to meet the service level requirements. On the contrary, γ > 1 suggests that extra capacity is needed by working overtime, introducing new LINACs, or rejecting some patients. We now compare the capacity allocation models without patient pooling (i.e. G = I) and with patient pooling G. A patient group g is said improving if patient pooling reduces the number of slots needed, i.e. N g iεig n i where n i is similar as N g but without patient pooling. It is said strictly improving if N g < iεig n i. Let γ I and γ G be the optimal LINAC utilization ratios of the two capacity allocation models. γ I γ G ( γ I > γ G ) if all patient groups are improving (strictly improving) as, for any solution s ij of the case without pooling, s gj = iεig s ij is a feasible solution for the case with pooling. C. Patient pooling This subsection addresses the patient pooling problem. The motivation of pooling two separate queues into a single queue and dedicated servers into shared servers is to avoid the situation of patient waiting in a queue while the other queue is empty and its servers idle. The negative effects include the degrading breaching probability of patients with smaller WTT and the increasing average waiting due to increasing service time variation. In this paper, we propose an original combinatorial optimization approach for patient pooling to improve the usage of slot severs while ensuring service level guarantee. Our main goal is to minimize the total number of slot servers needed and our second goal is to minimize the overall WTT breaching rate. The following additional notation is needed for patient pooling. Γ: set of all patient groups fulfilling assumption A5; π ig : actual breaching probability of type-i patients of group-g when N g slot servers are assigned to it; q ig : binary number equal to 1 if patient type-i belongs to group-g; ε: weighting factor of overall breaching rate. Decision variables of the patient pooling include: x g : binary variable equal to 1 if patient group g is selected. The patient pooling problem (PPP) consists in determining groups of patient types to pool in order to minimize the total number of slots needed and the overall breaching rate and meet all service level requirements. subject to min N g g Γ x g + ε λ i g Γ i I g π ig x g g Γ q ig x g =1, iεi (4) x g {0,1}, g Γ (5) Constraint (4) ensures that each patient type belongs to exactly one group and constraint (5) is a variable constraint. Note that N g in objective function and constraint (4) implicitly ensure all service level requirements, as N g is determined such that all service level requirements of patient types in g are met. The patient pooling problem can be solved by standard integer programing solvers with explicit enumeration of all possible patient groups. The major problem with the patient pooling model is the huge size of the set Γ that contains in the

6 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 6 worst case 2 I -1 patient groups. This motivates the heuristic algorithm proposed later. D. Case-mix optimization for a given patient pooling This subsection addresses the case-mix optimization under WTT constraints when there is no enough LINAC capacity. Additional defining elements include: λ i : least acceptance rate of type-i patients with λ i [0, λ i ], β i : unit reward of type-i patients. The unit rewards take into account the considerations of the hospital managers about the priorities in terms of the patientmix at a strategic level. It is usually related to ω i. Decisions include both LINAC capacity allocation decisions s gj and the following case-mix decisions: δ i : actual daily acceptance rate of type-i patients, which are assumed to arrive according to a Poisson process at rate δ i. The case-mix optimization problem (CMOP) consists in determining the acceptance rate of each patient type and allocating LINAC capacity to patient groups in order to maximize the total reward and meet service level requirements. subject to max iεi β i δ i jεjg s gj N g (δ g ), gεg (6) gεgj t gj s gj C j, jεj (7) λ i δ i λ i, iεi (8) s gj N, gεg, jεj (9) where δ g is the actual arrival rate vector of all patient types of group-g and N g (δ g ) is the minimal number of slot servers needed to meet all WTT requirements of group-g patient with arrival rate vector δ g. (6) ensures the service level requirements with respect to δ g. (7) is the LINAC capacity constraint. (8) is the acceptance rate constraint. (9) is the integrity constraint. IV. SOLUTION TECHNIQUES This section addresses the evaluation of the slot server requirements and solution techniques for capacity planning/allocation and patient pooling models of the previous section. The capacity allocation model for a given patient pooling can be easily solved by standard solvers and not addressed. The patient pooling model can be solved either directly by standard solvers or by the heuristic presented hereafter for realistic size. A linearization technique will be presented for the nonlinear case-mix optimization model. A. A heuristic for patient pooling Given the significant complexity of the patient partition problem, the direct solution of PPP with complete enumeration of all possible patient groups cannot be used for patient partition with realistic size. In this section, we propose a hill-climbing heuristic. It starts with the case without patient pooling, i.e. Γ = I and iteratively improves this initial patient pooling. At each iteration, it determines a set of patient group pairs to merge such that the PPP criterion reduces the most. More specifically, the local PPP optimization is characterized by the following notation: Γ : set of current patient groups; Π : set of pairs (g, g ) of patient groups in Γ that can be merged according to assumption A5 and such that (g, g ) Π implies (g, g ) Π if g g ; g g : new patient group obtained by merging patient groups g and g, i.e. group that contains all patient types in I g I g ; Decisions of the local PPP optimization include: z gg : binary variable equal to 1 if patient group pair (g, g ) are merged. z gg : binary variable equal to 1 if g is not merged to any others. The local PPP optimization consists in selecting the patient group pairs to merge in order to minimize the following: min (N g g + ε λ i π i,g g ) z gg (g,g ) Π subject to i I g g z gg (g,g ) Π and g g + z gg = 1, g Γ (10) z gg {0,1}, (g, g ) Π (11) where (10) ensures each group either be merged with another group or not. Pairwise merging heuristics for patient pooling 1. Initialize the set of current groups: Γ = I; 2. Evaluate N g and π ig for each current group-g; 3. Determine the set Π of possible pairwise merging; 4. Evaluate N g g and π i,g g for each pair (g, g ) Π such that g g ; 5. Solve the local PPP optimization; 6. Merge all patient group pairs (g, g ) with z gg = 1 and update Γ; 7. Repeat steps 3-6 if Γ has been changed and further merging is still possible. B. Case-mix optimization This subsection proposes a linearization of the nonlinear CMOP model that can then be solved by standard solvers. Due to the lack of closed-form expression of the nonlinear function N g (δ g ), it is approximated by a discrete function and the casemix optimization is then modified accordingly.

7 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 7 We assume that N g (δ g ) is defined only at a finite set g of discrete points δ gk with δ ik = λ i + m ik σ i where σ i = (λ i λ i ) K is the discretization step-size of δ i. We introduce the following auxiliary variable y gk : binary variable equal to 1 if δ g = δ gk. The case-mix optimization problem becomes: max ( β i δ ik ) y gk gεg subject to (7), (9) and kεδ g iεi g jεjg s gj kεδg N g (δ gk )y gk, gεg (12) kεδg y gk = 1, gεg (13) y gk {0,1}, gεg, kεδ g (14) Due to the complexity of taking into account patient pooling, the case-mix optimization is only tested for the case without patient pooling. In this case, δ g becomes a scalar, N g (δ g ) becomes a single variable function and the linearized model can be simplified accordingly. C. Performance evaluation of a patient group This subsection focuses on the performance evaluation of a given group g with a set I g of patient types served by slot servers of group-g according to the given priority rule. The main purpose is to determine the minimal number N g of slot servers needed and the actual breaching probability π ig of each patient type with N g slot servers. We first present some simple properties of the merged queue with pooled servers that help narrow the search space. First, as no patient leaves before the completion of her treatment sessions, WTT requirements can only be guaranteed under the stability of the system, i.e. relative server workload is less than 100% leading to: N g > i Ig λ i l i (15) Another important property is the monotonicity of the breaching probabilities π ig with respect to the number N g of slot servers. As the priority rule is either FIFO or a static priority rule, it can be shown that their treatment starting times are non-increasing in N g leading to the monotonicity of the breaching probabilities π ig. Thanks to the monotonicity, the determination of N g starts from some initial value and then searches either upward or downward till finding a minimal N g such that π ig α i for all patient types. The only remaining issue is the evaluation of the breaching probabilities for a given number N g of slot servers. Under the on-going assumptions, the system is similar to an M/D/n queue if there is one single patient type and an M/G/n queue if otherwise as the service time is the probability mixture of the session numbers of different patient types. The major difference with the classical M/G/n queue is the discrete time assumption of our system. Discrete event simulation is used to evaluate breaching probabilities as follows. At the beginning of each day, new patients arrive according to the Poisson rate λ g with λ g =. All patients wait in a queue and are served according i Ig λ i to the priority rule. Consider first the case of a FIFO rule. When a patient is selected to be served on a slot server, her type-i is randomly generated according to probability distribution λ i λ g, the service time is equal to l i and the breaching probability statistic of type-i is updated according to the waiting time of the patient and ω i. Consider now a static priority rule. The type-i of each incoming patient is randomly generated and she then joins the end of the queues corresponding to her priority class. When a slot server becomes available, it serves the first patient of the highest priority queue that is non empty. In order to reduce the variance and the error of the simulation, common patient arrival streams of different patient types are used for evaluation of different patient groups with different slot server numbers. The simulation starts with a warm up period of 10 4 days followed by a simulation of T days during which statistics are collected. Through some preliminary runs, we find T = 10 7 days to be a good balance between precision and computational effort. It is used in all numerical experiments. To further speed up the computation, we use an M/M/n approximation to determine the initial search point of N g. The M/M/n approximation has a Poisson arrival rate λ g and exponentially distributed service time with mean equal to μ 1 1 g = ( i λ i μ i ) λ g. The breaching probability of type-i patients can be approximated by the following existing results of M/M/n queues: P(W ω i ) = 1 C(n, a)e (nμ g λ g )ω i (16) C(n, a) = a n n!(1 ρ) a n 1 k k=0 k! + an n!(1 ρ) (17) a λ g μ g < n (18) where W is the steady-state waiting time of the M/M/n queue, a = λ g μ g the so-called offered load, ρ = a n the traffic intensity, C(n, a) the so-called Erlang C formula giving the waiting probability of an arrival patient. Formula (18) ensures the stability of the system. Table I compares the number of slot servers needed obtained the M/M/n approximation and the simulation for the case without patient pooling. The M/M/n approximation seems slightly overestimate the number of slot servers needed. It is used to set the starting N g for the case without patient pooling. Without patient pooling, the starting N g is iεig n i where n i is the least number of slots of each type. If the WTT requirements cannot be met with this slot number, then the

8 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 8 patient group is not improving and needs not to be considered in patient pooling optimization. Arrival rate TABLE I NUMBER OF SLOTS IN M/M/N APPROXIMATION VS SIMULATION Service rate max breach proba WTT (days) approx Simul % % V. NUMERICAL EXPERIMENTS This section presents numerical experiments to evaluate the performance of patient pooling heuristic and to study the properties of optimal solutions for capacity allocation, patient pooling and case-mix. All numerical experiments are performed on a workstation with processor Intel Core 16 with 32 CPU at 2.6 Ghz. All mathematical programming models are solved with commercial solver CPLEX12.2. A. Data setting We first present a base case from which the numerical experiments will be defined. It is derived from [3] considering 18 basic types of patients with different cancer locations (lung, prostate, breast, head, neck), treatment intents (palliative, radical) and emergency status. Two types of LINACs are considered: advanced LINACs (denoted ALINAC) can treat all patient types whereas regular LINACs (denoted RLINAC) cannot treat patient types 13 and 14. All LINACs work from 8:00 to 16:00 with a daily capacity of 480 minutes. The default priority rule is FIFO. Table II gives other patient information where arrival rate (λ i ) and WTT (ω i ) are measured in days, durations (t ij ) are given in minutes on ALINAC/RLINAC, slot nb is the minimal number n i of slot servers needed for each patient type without pooling, BP is the breaching probability. The total arrival rate is 8.25 patients per day. The default maximal breaching probability α i is 5%. The default weighting factor of total breaching rate in patient pooling is 1 as our main objective is to minimize the number of slots. According to assumption A5, patient types 6 and 14 with session duration of 24 minutes cannot be merged with other patient types with session duration of 12 minutes. Patient type 13 can be merged with others if only ALINACs are considered and it should be kept alone if both ALINACs and RLINACs are considered. B. Is patient pooling always beneficial Let us consider the simulation results of pooling patient types 1, 3, 4 and 8 in Table II. Pooling patients 1 and 3 (similar WTT and service time) needs 3 slots and saves 1 slot. Pooling patients 1 and 4 (different WTT and service time) requires 28 slots, 2 slots more than without pooling. Pooling patients 1 and 8 (similar service time but different WTT) needs 9 slots as without pooling but leads to higher BP (0.036) for type 1 patients and lower BP (0.000) for type 8 patients. To conclude, whether patient pooling is beneficial is not a trivial issue and it depends on their service times (i.e. number of sessions), their WTT and the priority rules. TABLE II PATIENT INFORMATION type session nb arrival rate WTT duration slot nb / / / / / / / / / / / / / / / / / / C. Efficiency of the patient pooling heuristic This subsection compares the pairwise merging heuristic and the exact method by exhaustive enumeration and standard MIP solvers. The size of the problem instances of the base case being too large for the exact method, the comparison is carried out on 10 problem instances with only ALINACs and 10 patient types sampled from the initial 18 types. The list of patient types sampled for each instance is given in Table III. Table IV gives for each instance and for each method (Opt=Optimal, Heu=Heuristic) the total number of slots, the total breaching rate and the patient pooling. For each group of patients, the number of slots needed is given in parentheses. The total number of slots is exactly the same for both methods except for problem instance S8 with a deviation of 1 slot. The total breaching rate deviates by less than and is the same for four instances which is about 1.2% of the average daily arrival rate. The relative deviation of the criterion value is 1.6% for problem instance S8 and less than 0.1% for all other instances. The heuristic algorithm is at least 10 times faster than the exact method. More specifically, the (min, ave, max) of the CPU time in minutes is (389, 942, 1475) with the exact method and (31, 64, 95) with the heuristic. Given the excellent performance of the pairwise merging heuristic, it is used in all remaining numerical experiments. BP

9 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 9 TABLE III PROBLEM INSTANCES OF 10 PATIENT TYPES S1 S2 S3 S4 S5 S6 S7 S8 S9 S TABLE IV PAIRWISE MERGED POOLING VS OPTIMAL PATIENT POOLING slots BP pooling Opt {14(11)}, {1,2,3,7(4)},{8,9,13,16,18(33)} Heu {14(11)}, {1,2,3,7(4)}, {8,9,13,16,18(33)} Opt , {2,3,10(3)}, {5,8,9,16,18(38)} Heu , {2,3,8,10(10)}, {5,9,16,18(31)} Opt {6(8)}, {2,7,12(2)},{4,8,9,10,11,17(46)} Heu {6(8)}, {2,7(2)}, {4,8,9,10,11,12,17(46)} Opt {14(11)}, {2,3,7(3)}, {4,5,10,13,15,16(66)} Heu {14(11)}, {2,3,7,10,13(12)}, {4,5,15,16(57)} Opt {2,3,12(2)}, {4,5,8,9,11,15,18(63)} Heu {2,3,12(2)}, {4,5,8,9,11,15,18(63)} Opt {6(8)}, {4,5,8,15(53)},{1,2,3,7,12 (4)} Heu {6(8)}, {4,5,8,12,15(53)}, {1,2,3,7 (4)} Opt {6(8)}, {1,7(3)}, {9,10,13,15,16,17,18(44)} Heu {6(8)}, {1,7(3)}, {9,10,13,15,16,17,18(44)} Opt {14(11)}, {3,4,12,16(31)}, {1,2,5,10,13(17)} Heu {14(11)}, {3,4,5,1012,13,16(47)}, {1,2 (2)} Opt {6(8)}, {11,16,17,18(22)}, {1,3,7,9,13 (17)} Heu {6(8)}, {16,17,18(19)}, {1,3,7,9,11,13 (20)} Opt , {2,3,10(4)}, {5,8,9,16,18(37)} Heu , {2,3,10(4)}, {5,8,9,16,18(37)} D. Capacity allocation for the base case The subsection is the detailed presentation of the capacity allocation, patient pooling and case-mix optimization for the base case defined in Section VA. Four scenarios of LINAC mix are considered: scenario 1 with 4 ALINAC, scenario 2 with 3 ALINAC and 1 RLINAC, scenario 3 with 2 ALINAC and 2 RLINAC, and scenario 4 with 4 RLINACs. Capacity allocation without pooling: From Table II, the total number of slots needed in this case is 125. Table V gives the LINAC utilization when overtime (OT) is not needed and OT is given when a LINAC is overused. Capacity allocation with patient pooling: The pairwise merging heuristic leads to the patient groups of Table VI with significant reduction in total number of slots needed from 125 to 110 or 111 depending on the scenario. LINAC capacity allocation for these patient groups is given in Table VII where groups 4 and 5 can only be assigned to ALINACs. Table VIII gives the LINAC utilization. TABLE V LINAC UTILIZATION WITHOUT PATIENT POOLING Scenario 1 Scenario 2 Scenario 3 Scenario 4 LINAC1 A 90% R 93.75% R 100% R OT 30min LINAC2 A 90% A 95% R 100% R OT 30min LINAC3 A 90% A 95% A 92.5% R 100% LINAC4 A 90% A 95% A 100% A OT 24min Scenario Slot AAAA 110 Others 111 TABLE VI PATIENT POOLING Patient groups {1,2,3,7(4)}, {4,5,8,9,10,11,12,13,15,16,17,18(88)}, {1,2,3,7(4)}, {4,5,8,9,10,11,12,15,16,17,18(80)}, {6(8)}, {13(8)}, {14(11)} TABLE VII CAPACITY ALLOCATION FOR PATIENT GROUPS Scenario 1 Scenario 2 Scenario 3 Scenario 4 gr A A A A R A A A R R A A R R R A TABLE VIII LINAC UTILIZATION WITH PATIENT POOLING Scenario 1 Scenario 2 Scenario 3 Scenario 4 LINAC1 A 80% R 84.4% R 90.6% R 96.9% LINAC2 A 80% A 87.5 R 90.6% R 96.9% LINAC3 A 80% A 85% A 90% R 96.9% LINAC4 A 80% A 85% A 90% A 95% The following observations can be made. First significant improvement of the LINAC capacity utilization is achieved and the actual LINAC capacity is now enough to meet all WTT requirements. Patient types except 6, 13 and 14 can be merged into a single group according to assumption A5 and are grouped as follows. Group 1 contains all patient types with short WTT. Group 2 contains all patient types with WTT=40 with those needing over 30 sessions and those needing only 1 session. The characteristics of patient pooling will be addressed in details later. Another interesting observation is the pooling of patient types with very low capacity requirement such as types 7 and 12. They can be put in different groups without changing the total numbers of slots but slight change of breaching rate. Here,

10 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 10 type 7 is put in group 1 with other types of short WTT. Similar observation can be made for type 12 in Table IV. Case-mix optimization without pooling: We assume that only 3 LINACs are available. Three scenarios AAA, RAA and RRA are considered and the capacity is not enough in all scenarios. The least acceptance rate is λ i = 0. The weighting factors β i given in Table IX are set as follows: 8 if WTT = 5, 4 if WTT = 10, and 1 or 2 if WTT = 40. Table IX gives the percentage of real demand accepted (called coverage) and in parenthesis the number of missing slots for full coverage. For all scenarios, all LINAC are fully used. All patient types with WTT = 5 are fully accepted. TABLE IX PATIENT COVERAGE IN CASE-MIX OPTIMIZATION type weight session nb AAA RAA RRA % 100% 100% % 100% 100% % 100% 100% % 100% 100% % 100% 100% % 100% 100% % 100% 100% % 98.2% (1) 98.2% (1) % 100% 100% % 100% 100% % 100% 100% % 50% (5) 0% (11) % (12) 0% (12) 0% (12) % (7) 0% (7) 0% (7) % (3) 0% (3) 0% (3) % 88.9% (1) 75.6% (2) % 100% 100% % (1) 95.2% (1) 95.2% (1) E. Sensitivity analysis of patient pooling This subsection addresses the sensitivity of patient pooling with respect to three key parameters (WTT ω i, maximal breaching probability α i, weighting factor ε of breaching rate). Only ALINACs are considered. As a result, all patient types except types 6 and 14 can be merged into a single group. Sensitivity with respect to WTT. Table X gives for each case the total number of slots with and without pooling and the patient groups. (i) The number of slots needed decreases as the WTT increases. (ii) Patient types with different WTTs can be merged into a single group. Types 6 and 14 are grouped in a separate group 3 even if they have different WTTs. (iii) The variation of WTT seems to have a significant impact on the patient pooling. For the last three cases with identical WTT, all patient types except 6 and 14 are grouped into a single group. In the first three cases with distinct WTT, group 1 contains all patient types (1, 2, 3) with the shortest WTT whereas group 2 contains those (15, 16, 17, 18, 9, 11) with the longest WTT and largest number of sessions. The other patient types, especially those with small session numbers, switch between these two groups depending on the WTT. Sensitivity with respect to the increase of the maximal breaching probability (Table XI). (i) The total number of slots decreases. (ii) The number of patient groups decreases. Pooling all patients into a single group is unlikely to be beneficial when the WTT requirement is high ( α i = 3%), whereas it is likely when the WTT requirement is low. Sensitivity with respect to the weighting factor of breaching rate ε. As our main objective is to minimize the total number of slots, the weighting factor is small with ε = 1. As ε increases, the total number of slots becomes less important whereas the total breaching rate becomes more important. For this reason, as ε increases, the total number of slots increases, the number of patient groups increases, and the total breaching rate drops from to with respect to 0.1 for the case without pooling. TABLE X PATIENT POOLING VS WTT WTT slots Patient groups 0.8Base 110/129 Base 110/ Base 108/124 {1,2,3,7,8,10,12,13(20)}, {4,5,9,11,15,16,17,18(72)}, {1,2,3,7(4)}, {4,5,8,9,10,11,12,13,15,16,17,18(88)}, {1,2,3,7,8,10,12,13(19)}, {4,5,9,11,15,16,17,18(72)}, {6,14(17)} 5 112/151 {1-5,7-13,15-18(93)}, {6,14(19)} /137 {1-5,7-13,15-18(91)}, /121 {1-5,7-13,15-18(89)}, {6,14(16)} TABLE XI PATIENT POOLING VS BREACHING PROBABILITY max BP slots Patient groups 3% 110/128 5% 110/125 10% 108/121 {1,2,3,7 (4)}, {8,10,12,13 (23)}, {4,5,9,11,15,16,17,18(65)}, {1,2,3,7(4)}, {4,5,8,9,10,11,12,13,15,16,17,18(88)}, {1,2,3,7,8,10,12,13(19)}, {4,5,9,11,15,16,17,18(72)}, {6,14(17)} 15% 108/121 {1-5,7-13,15-18(91)}, {6,14(17)} TABLE XII PATIENT POOLING VS WEIGHTING FACTOR OF BREACHING RATE weight slots Patient groups {1,2,3,7(4)}, {4,5,8,9,10,11,12,13,15,16,17,18(88)}, {1,2,3,7(4)}, {4,5,8,9,10,11,12,13,15,16,17,18(88)}, {1,3,5,17(23)},{2,7 (2)}, {4,8,9,10,11,12,13,15,16,18(71)}, {1,15(16)}, {2,3,7(3)}, {4,17(30)}, {5,9(19)}, {8,13(15)},{11(4)},{10,12(2)},{16,18(14)}, WTT versus capacity requirement of a given group. This analysis is motivated by strong pressures of hospital managers to reduce the capacity requirement by extending the WTT. For this purpose, three patient groups G1 = {4, 5, 6, 8, 10, 11, 12,

11 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 11 13, 15, 16, 17, 18}, G2 = {1, 2, 3, 7}, G3 = {6, 14} are considered. Table XIII shows the number of slots needed for each patient group and for different WTT where xbase implies the multiplication of the base WTT by x. Diminishing benefit of increased WTT is observed. Beyond the regular WTT, the capacity saving is at the cost of unreasonably low service level. For example, double the base WTT saves no more than one slot server. TABLE XIII NUMBER OF SLOTS FOR A GIVEN PATIENT GROUP VS WTT 0.2Base 0.4Base 0.6Base 0.8Base Base 2Base 10Base 30Base G G G F. Impact of priority rule This section addresses the impact of the priority rule on capacity requirement. Two priority rules are considered: the commonly used FIFO rule and a new policy termed NPP. NPP serves first patients with the shortest WTT and patients with the smallest session number in case of equal WTT. We first address the capacity requirement for six patient groups of two patient types whose parameters (arrival rate, service rate, WTT) are given in Table XIV. We can see that which of FIFO and NPP performs better depends on the combination of arrival rate, service rate and WTT. We then address the impact of priority rule on patient grouping. Table XV shows the optimal patient grouping with both policies for the base case. While the overall capacity requirement is the same with 110 slots, patient groupings are different. NPP merges into a single group all patient types except 6, 14. While FIFO needs one more slot for the patient group {6, 14}, it saves one slot for other patients by finer partition of patients. Policy slots TABLE XIV NUMBER OF SLOTS VS SCHEDULING POLICIES λ i /1.43 /1.43 /0.99 /0.19 /1.43 /1.43 μ i 5/16 16/5 16/5 16/5 16/16 5/16 WTT 5/40 5/6 5/6 5/6 5/40 5/5 NPP FIFO TABLE XV PATIENT GROUPING VS SCHEDULING POLICIES Patient groups NPP 110 {1,2,3,4,5,7,8,9,10,11,12,13,15,16,17,18(93)}, {6,14(17)} FIFO 110 {1,2,3,7(4)}, {4,5,8,9,10,11,12,13,15,16,17,18(88)}, In general, FIFO avoids merging in the same group patients with significant different WTT as all patients in the same group have the same service level with FIFO. With static priority in NPP, a patient type i with slightly smaller WTT or service time than another patient type j has much lower service level that patient type j, leading to higher capacity requirement. Given above, even though FIFO is not always optimal, it can serve as a good basis for capacity allocation at the strategic and tactical levels. It also makes the capacity allocation model more tractable. VI. CONCLUSION In this paper, we have addressed capacity allocation for radiotherapy treatment with heterogeneous LINACs for multiple patient types having different treatment protocols and different WTT. The main contribution is a queueing and patient pooling approach for LINAC capacity allocation. A queueing framework was proposed by considering time slots of LINAC as servers. This original queueing perspective allows nice modeling of treatment protocols with single-class queueing models that, otherwise, would lead to complicated multi-class queueing systems with re-entrance. It greatly simplifies the mathematical formulation of the LINAC capacity planning problems. Mixed integer programming models are given for the capacity allocation and case-mix optimization problems. The queueing framework was extended to pool patient types into groups served by the same slot servers to efficiently reduce the total number of slot servers. A mathematical programming model and a pairwise merging heuristic were proposed for patient pooling optimization. This research can be extended in several directions. The breaching probability evaluation of a multi-class multi-server queue with multiple WTT is a challenging open issue but important to speed up our approach. Joint patient pooling and case-mix optimization is highly combinatorial and requires efficient solution techniques. Joint optimization of the service level and case-mix is another interesting direction. Finally the real-time patient admission control of multiple patient types with heterogeneous WTT is the next step of improving radiotherapy operations. ACKNOWLEDGEMENT The authors are grateful to physicians and nurses of the Radiotherapy Unit headed by Dr. Chen Xu of the Ruijin Hospital, Shanghai, China for discussions and information. REFERENCES [1]S.Petrovic, N.Mishra and S.Sundar, A novel case based reasoning approach to radiotherapy planning, Expert Systems with Applications, 38(9): , [2] T.Kapamara, K.Sheibani, D. Petrovic, et al, A simulation of a radiotherapy treatment system: A case strudy of local cancer center, Proc. ORP3 Conference, Guimaraes, Portugal, 29-35, [3] M. S. Thomsen and O. Nørrevang, "A model for managing patient booking in a radiotherapy department with differentiated waiting times," Acta oncologica, 48(2): , [4] Z. Chen, W. King, R. Pearcey, M. Kerba, and W. Mackillop. The relationship between waiting time for radiotherapy and clinical outcomes: a systematic review of the literature, Radiotherapy and Oncology, 87 (1): 3 16, 2008.

12 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 12 [5] A. Sauré, J. Patrick, S. Tyldesley, and M. L. Puterman, "Dynamic multiappointment patient scheduling for radiation therapy," European Journal of Operational Research, 223(2): , [6] S. Petrovic and P. Leite-Rocha, "Constructive approaches to radiotherapy scheduling," In Proc. World congress on engineering and computer science (WCECS'08), , [7] W.Whitt, "Partitioning Customers into Service Groups," Management Science, 45(11): , [8] E.Tekin, W.J.Hopp, and M.P.Van Oyen, Pooling Strategies for Call Center agent Cross-training, IIE Transactions, 41(6): , [9] P.E. Joustra, E. van der Sluis,and N.M. van Dijk, To pool or not to pool in hospitals: A theoretical and practical comparison for a radiotherapy outpatient department, Annals of Operations Research,178:77-89, [10] S. Petrovic, W. Leung, X. Song, and S. Sundar, "Algorithms for radiotherapy treatment booking," in Proc. Workshop UK Planning and Scheduling Special Interest Group (PlanSIG), Nottingham, UK, [11] D. Conforti, F. Guerriero, and R. Guido, "Optimization models for radiotherapy patient scheduling," 4OR, 6(3): , [12] D. Conforti, F. Guerriero, R. Guido, and. M. Veltri, "An optimal decision-making approach for the management of radiotherapy patients," OR Spectrum, 33(1): , [13] D. Conforti, F. Guerriero, and R. Guido, "Non-block scheduling with priority for radiotherapy treatments," European Journal of Operational Research, 201(1): , [14] E.K. Burke, P. Leite-Rocha, and S. Petrovic, "An Integer Linear Programming Model for the Radiotherapy Treatment Scheduling Problem," arxiv preprint arxiv: , [15] L.V. Green, S. Savin, and B. Wang, Managing patient service in a diagnosis medical facility, Operations Research, 54(1):11-25, [16] N. Geng, X. Xie, V. Augusto, and Z. Jiang, "A Monte Carlo Optimization and Dynamic Programming Approach for Managing MRI Examinations of Stroke Patients," IEEE Transactions on Automatic Control, 56(11): 2515, 2529, [17] E.P.C Kao and G.G. Tung, "Bed Allocation in a Public Health Care Delivery System," Management Science, 27(5): , [18] E.N. Weiss, Models for determining estimated start times and case orderings in hospital operating rooms, IIE Transactions, 22(2): , [19] S. Li, X. Xie, and N. Geng, A Queuing Approach for Radiotherapy Treatment Capacity Planning, Proc. IEEE Conf. Auto. Sci. & Engr. (CASE 2014), to appear. Xiaolan Xie (M 93 SM 10-F 15) received his Ph.D degree from the University of Nancy I, Nancy, France, in 1989, and the Habilitation àdiriger des Recherches degree from the University of Metz, France, in Currently, he is a Distinguished Professor of industrial engineering, the Head of the Department of Healthcare Engineering of the Center for Biomedical and Healthcare Engineering and the Head of IEOR team of CNRS UMR 6158 LIMOS, Ecole Nationale Supérieure des Mines (ENSMSE), Saint Etienne, France. He is also a Chair Professor and Director of the Center for healthcare engineering at the Shanghai Jiao Tong University, China. Before Joining ENSMSE, he was a Research Director at the Institut National de Recherche en Informatique et en Automatique (INRIA) from 2002 to 2005, a Full Professor at Ecole Nationale d Ingénieurs de Metz from 1999 to 2002, and a Senior Research Scientist at INRIA from 1990 to His research interests include design, planning and scheduling, supply chain optimization, and performance evaluation, of healthcare and manufacturing systems. He is author/coauthor of over 250 publications including over 90 journal articles and six books. He has rich industrial application experiences with European industries. He is PI for various national and international projects including ANR-TECSAN HOST on management of winter epidemics, NSF China key project on planning and optimization of health care resources, French Labex IMOBS3 project on home health cares, FP6-IST6 IWARD on swarm robots for health services, FP6- NoE I*PROMS on intelligent machines and production systems, the FP5- GROWTH-ONE project for the strategic design of supply chain networks, the FP5- GRWOTH thematic network TNEE on extended enterprises. Dr. Xie has been an associate editor for International Journal for Production Research, IEEE Transactions on Automation Science & Engineering, IEEE Transaction on Automatic Control, IEEE Transactions on Robotics & Automation and has been on the (conference) editorial board of the IEEE Robotics and Automation Society, Health Systems, and Int. J. Simulation & Process Modelling. He has a Guest Editor of various special issues on healthcare engineering and manufacturing systems. He is general chair of ORAHS 2007, IPC chair of the IEEE Workshop on Health Care Management WHCM 2010 and IPC member for many other conferences. He is the corresponding co-chair of the IEEE RAS Technical Committee on Automation in Healthcare Management. Siqiao Li received the B.S. degree in Logistics Engineering from Shandong University, Jinan, China, in Currently, she is a Ph.D. student in the Department of Industrial Engineering and Management, Shanghai Jiao Tong University, Shanghai, China. Her research interests include planning and scheduling of healthcare systems. Na Geng received her Ph.D. in Industrial Engineering from both Ecole Nationale Superieure des Mines de Saint-Etienne (ENSM.SE), France and Shanghai Jiao Tong University (SJTU), China, in Currently, she is an associate professor in the Department of Industrial Engineering and Management, SJTU, Shanghai, China. Her research interests include capacity planning and control, scheduling etc. She is the author/coauthor of over 30 papers. She is the PI of two NSF of China funded projects.